Monge-Amp\`ere equation, hyperk\"ahler structure and adapted complex structure

Abstract

In the tangent bundle of $(M,g)$, it is well-known that the Monge-Amp\`ere equation $(\partial\bar\partial \sqrt\rho)^n=0$ has the asymptotic expansion $ \rho(x+iy)=\sum_{ij} g_{ij} (x) y_{i} y_{j} + O(y^4)$ near $M$. Those 4th order terms are made explicit in this article: $$\rho(x+iy)=\sum_{i}y_{i}^2-\frac 13\sum_{pqij} R_{i p j q}(0)x_p x_q y_{i}y_{j}+O(5).$$ At $M$, sectional curvatures of the K\"ahler metric $2i\partial\bar\partial\rho$ can be computed. This has enabled us to find a family of K\"ahler manifolds whose tangent bundles have admitted complete hyperk\"ahler structures whereas the adapted complex structure can only be partially defined on the tangent bundles. In these cases, the study of the adapted complex structure is equivalent to the study of some gauge transformations on the baby Nahm's equation $\dot T_1+[T_0,T_1]=0.$